Centre of mass of solid cone

  • #1
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Homework Statement



Find the centre of mass of solid cone.

Homework Equations



$$y_{cm}=\frac{1}{M}\int_0^Hydm$$

The Attempt at a Solution


First I took thin disks. I got the answer when I assumed its thickness to be dy but then dysecθ would be more accurate if half angle of cone is θ since dysecθ gives the length of the slanted part, while dy would give a cylinder.
20150329_112303-1.jpg

Now I get wrong answer.
 

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Answers and Replies

  • #2
ehild
Homework Helper
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1,912

Homework Statement



Find the centre of mass of solid cone.

Homework Equations



$$y_{cm}=\frac{1}{M}\int_0^Hydm$$

The Attempt at a Solution


First I took thin disks. I got the answer when I assumed its thickness to be dy but then dysecθ would be more accurate if half angle of cone is θ since dysecθ gives the length of the slanted part, while dy would give a cylinder.
But the volume of a truncated cone is (h pi/3 )(r12+r1r2+r22) where r1 and r2 are the radii of the base plates and h is the height. In case of the volume element, r1≈r2=r and h=dy, so the volume element is that of a disk with radius r and height dy.

Also, you have to use density times Ady instead of dm in the integral.
 
Last edited:
  • #3
528
33
But the volume of a truncated cone is (h pi/3 )(r12+r1r2+r22) where r1 and r2 are the radii of the base plates and h is the height. In case of the volume element, r1≈r2=r and h=dy, so the volume element is that of a disk with radius r and height dy.

Also, you have to use density times dy instead of dm in the integral.
OK. But whats wrong in dysecθ?
 
  • #4
ehild
Homework Helper
15,543
1,912
Nothing is wrong with it if you do not use it instead of dy.
 

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